Properties

Degree 2
Conductor $ 3 \cdot 17 \cdot 157 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.514·2-s − 3-s − 1.73·4-s − 2.23·5-s + 0.514·6-s + 1.18·7-s + 1.92·8-s + 9-s + 1.14·10-s − 3.76·11-s + 1.73·12-s − 1.73·13-s − 0.611·14-s + 2.23·15-s + 2.48·16-s − 17-s − 0.514·18-s + 1.01·19-s + 3.87·20-s − 1.18·21-s + 1.93·22-s + 2.15·23-s − 1.92·24-s − 0.0244·25-s + 0.890·26-s − 27-s − 2.06·28-s + ⋯
L(s)  = 1  − 0.363·2-s − 0.577·3-s − 0.867·4-s − 0.997·5-s + 0.209·6-s + 0.449·7-s + 0.679·8-s + 0.333·9-s + 0.362·10-s − 1.13·11-s + 0.501·12-s − 0.480·13-s − 0.163·14-s + 0.575·15-s + 0.620·16-s − 0.242·17-s − 0.121·18-s + 0.233·19-s + 0.865·20-s − 0.259·21-s + 0.412·22-s + 0.449·23-s − 0.392·24-s − 0.00488·25-s + 0.174·26-s − 0.192·27-s − 0.390·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8007\)    =    \(3 \cdot 17 \cdot 157\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{8007} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 8007,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;17,\;157\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;17,\;157\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + T \)
17 \( 1 + T \)
157 \( 1 + T \)
good2 \( 1 + 0.514T + 2T^{2} \)
5 \( 1 + 2.23T + 5T^{2} \)
7 \( 1 - 1.18T + 7T^{2} \)
11 \( 1 + 3.76T + 11T^{2} \)
13 \( 1 + 1.73T + 13T^{2} \)
19 \( 1 - 1.01T + 19T^{2} \)
23 \( 1 - 2.15T + 23T^{2} \)
29 \( 1 + 1.45T + 29T^{2} \)
31 \( 1 + 4.31T + 31T^{2} \)
37 \( 1 + 1.11T + 37T^{2} \)
41 \( 1 - 2.21T + 41T^{2} \)
43 \( 1 - 9.50T + 43T^{2} \)
47 \( 1 - 5.01T + 47T^{2} \)
53 \( 1 - 13.2T + 53T^{2} \)
59 \( 1 - 3.08T + 59T^{2} \)
61 \( 1 + 9.73T + 61T^{2} \)
67 \( 1 + 1.07T + 67T^{2} \)
71 \( 1 + 13.7T + 71T^{2} \)
73 \( 1 - 5.31T + 73T^{2} \)
79 \( 1 + 9.97T + 79T^{2} \)
83 \( 1 - 14.0T + 83T^{2} \)
89 \( 1 - 7.10T + 89T^{2} \)
97 \( 1 - 1.44T + 97T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.49219327273848737934682244506, −7.22982146230811808303541306661, −5.88754572556508276834784361537, −5.31983425086625004648577228500, −4.63825159719728571508274998514, −4.13441977460339870640589252968, −3.23966916393583817802406794566, −2.09956386686401479756470582376, −0.831861063167652053161786471456, 0, 0.831861063167652053161786471456, 2.09956386686401479756470582376, 3.23966916393583817802406794566, 4.13441977460339870640589252968, 4.63825159719728571508274998514, 5.31983425086625004648577228500, 5.88754572556508276834784361537, 7.22982146230811808303541306661, 7.49219327273848737934682244506

Graph of the $Z$-function along the critical line